82 research outputs found
Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces
This paper is the third of a series on Hamiltonian stationary Lagrangian
surfaces. We present here the most general theory, valid for any Hermitian
symmetric target space. Using well-chosen moving frame formalism, we show that
the equations are equivalent to an integrable system, generalizing the C^2
subcase analyzed in the first article (arXiv:math.DG/0009202). This system
shares many features with the harmonic map equation of surfaces into symmetric
spaces, allowing us to develop a theory close to Dorfmeister, Pedit and Wu's,
including for instance a Weierstrass-type representation. Notice that this
article encompasses the article mentioned above, although much fewer details
will be given on that particular flat case
A canonical structure on the tangent bundle of a pseudo- or para-K\"ahler manifold
It is a classical fact that the cotangent bundle T^* \M of a differentiable
manifold \M enjoys a canonical symplectic form . If
(\M,\j,g,\omega) is a pseudo-K\"ahler or para-K\"ahler -dimensional
manifold, we prove that the tangent bundle T\M also enjoys a natural
pseudo-K\"ahler or para-K\"ahler structure (\J,\G,\Omega), where is
the pull-back by of and \G is a pseudo-Riemannian metric with
neutral signature . We investigate the curvature properties of the
pair (\J,\G) and prove that: \G is scalar-flat, is not Einstein unless
is flat, has nonpositive (resp.\ nonnegative) Ricci curvature if and only if
has nonpositive (resp.\ nonnegative) Ricci curvature as well, and is
locally conformally flat if and only if and has constant curvature,
or and is flat. We also check that (i) the holomorphic sectional
curvature of (\J,\G) is not constant unless is flat, and (ii) in
case, that \G is never anti-self-dual, unless conformally flat.Comment: Clarified the statements on the cotangent bundle. Corrected various
typo
Ricci curvature on polyhedral surfaces via optimal transportation
The problem of defining correctly geometric objects such as the curvature is
a hard one in discrete geometry. In 2009, Ollivier defined a notion of
curvature applicable to a wide category of measured metric spaces, in
particular to graphs. He named it coarse Ricci curvature because it coincides,
up to some given factor, with the classical Ricci curvature, when the space is
a smooth manifold. Lin, Lu & Yau, Jost & Liu have used and extended this notion
for graphs giving estimates for the curvature and hence the diameter, in terms
of the combinatorics. In this paper, we describe a method for computing the
coarse Ricci curvature and give sharper results, in the specific but crucial
case of polyhedral surfaces
Discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. A discrete Lawson correspondence
The main result of this paper is a discrete Lawson correspondence between
discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a
correspondence between two discrete isothermic surfaces. We show that this
correspondence is an isometry in the following sense: it preserves the metric
coefficients introduced previously by Bobenko and Suris for isothermic nets.
Exactly as in the smooth case, this is a correspondence between nets with the
same Lax matrices, and the immersion formulas also coincide with the smooth
case.Comment: 13 page
Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface
Given an oriented Riemannian surface , its tangent bundle
enjoys a natural pseudo-K\"{a}hler structure, that is the combination
of a complex structure \J, a pseudo-metric \G with neutral signature and a
symplectic structure \Om. We give a local classification of those surfaces of
which are both Lagrangian with respect to \Om and minimal with
respect to \G. We first show that if is non-flat, the only such surfaces
are affine normal bundles over geodesics. In the flat case there is, in
contrast, a large set of Lagrangian minimal surfaces, which is described
explicitly. As an application, we show that motions of surfaces in or
induce Hamiltonian motions of their normal congruences, which are
Lagrangian surfaces in or T \H^2 respectively. We relate the area of
the congruence to a second-order functional
on the original surface.Comment: 22 pages, typos corrected, results streamline
Courbure discrète : théorie et applications
International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor
A rigidity theorem for Riemann's minimal surfaces
International audienceWe describe first the analytic structure of Riemann's examples of singly-periodic minimal surfaces; we also characterize them as extensions of minimal annuli bounded by parallel straight lines between parallel planes. We then prove their uniqueness as solutions of the perturbed problem of a punctured annulus, and we present standard methods for determining finite total curvature periodic minimal surfaces and solving the period problems
Bijective rigid motions of the 2D Cartesian grid
International audienceRigid motions are fundamental operations in image processing. While they are bijective and isometric in R^2, they lose these properties when digitized in Z^2. To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on Z^2, initially proposed by Nouvel and Rémila for rotations on Z^2. This allows us to study bijective rigid motions on Z^2, and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of Z^2 is bijective
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